$A$ long solenoid with $10.0 \text{ turns/cm}$ and a radius of $8 \text{ cm}$ carries a current of $7 \text{ mA}$. $A$ current-carrying straight conductor is located along the central axis of the solenoid. If the direction of the resulting magnetic field is $60^{\circ}$ to the axial direction at a point $5 \text{ cm}$ from the axis of the solenoid along the radial direction, then the current in the conductor is. [Take $\sqrt{2}=1.4, \sqrt{3}=1.7$] (in $\text{ A}$)

The magnetic field at the centre of a long solenoid having $400$ turns per unit length and carrying a current $i$ is $6.24 \times 10^{-2} \,T$. The magnetic field at the centre of another long solenoid having $200$ turns per unit length and carrying a current $\frac{i}{2}$ is

Consider a circular current-carrying loop of radius $R$ in the $x-y$ plane with its centre at the origin. Consider the line integral $\Im(L) = \left| \int_{-L}^{L} \vec{B} \cdot d\vec{l} \right|$ taken along the $z$-axis.
$(a)$ Show that $\Im(L)$ monotonically increases with $L$.
$(b)$ Use an appropriate Amperian loop to show that $\Im(\infty) = \mu_0 I$,where $I$ is the current in the wire.
$(c)$ Verify this result directly.
$(d)$ Suppose we replace the circular coil with a square coil of side $R$ carrying the same current $I$. What can you say about $\Im(L)$ and $\Im(\infty)$?

$A$ particle of mass $1 \times 10^{-27} \ kg$ and charge $1 \times 10^{-16} \ C$ enters a uniform magnetic field within a solenoid at a speed of $1000 \ m/s$. The velocity vector makes an angle of $60^{\circ}$ with the axis of the solenoid. The solenoid has $5000$ turns along its length $L$ and carries a current of $5 \ A$. The number of revolutions the particle makes along the helical path within the solenoid by the time it emerges from the solenoid's opposite end is:

$A$ long solenoid has $200$ turns per $cm$ and carries a current $i$. The magnetic field at its centre is $6.28 \times 10^{-2} \ Wb/m^2$. Another long solenoid has $100$ turns per $cm$ and it carries a current $i/3$. The value of the magnetic field at its centre is:

An infinitely long hollow conducting cylinder with radius $R$ carries a uniform current along its surface. Choose the correct representation of magnetic field $(B)$ as a function of radial distance $(r)$ from the axis of the cylinder.